
    \hj                     V   S SK JrJrJrJrJrJrJrJrJ	r	J
r
JrJrJrJrJrJrJr  S SKJrJr  S SKJrJr  S SKJr  S SKJrJrJrJr  S SKJ r   S SK!J"r"J#r#  S SK$J%r%  S S	K&J'r'J(r(  S S
K)J*r*J+r+J,r,  S SK-J.r.  S SK/J0r0J1r1J2r2J3r3  S SK4J5r5J6r6J7r7J8r8  S SK9J:r:  S SK;J<r<J=r=  S SK>J?r?  S SK@JArA  S SKBJCrC  S SKDJErE  S SKFJGrG  S SKHJIrIJJrJJKrKJLrL  S SKMJNrNJOrOJPrPJQrQJRrRJSrS  \#" S5      u  rTrUrVS rWS rXS rY\J\IS 5       5       rZS r[S r\\JS 5       r]\JS 5       r^\JS  5       r_S! r`S" raS# rbS$ rcS% rdS& reS' rfg())    )mellin_transforminverse_mellin_transformfourier_transforminverse_fourier_transformsine_transforminverse_sine_transformcosine_transforminverse_cosine_transformhankel_transforminverse_hankel_transformFourierTransformSineTransformCosineTransformInverseFourierTransformInverseSineTransformInverseCosineTransformIntegralTransformError)laplace_transforminverse_laplace_transform)Function
expand_mul)
EulerGamma)IRationaloopi)S)Symbolsymbols)	factorial)re
unpolarify)exp	exp_polarlog)sqrt)atancossintan)besselibesseljbesselkbessely)	Heaviside)erfexpint)gamma)meijerg)	gammasimp)hyperexpand)trigsimp)XFAILslowskipraises)xsabcdznu beta rhoc                     SSK Jn   [        S5      n[        U" [        5      [        [
        5      U " U" [        5      [        [
        5      :X  d   e[        U" [        5      [        [        * 5      -   [        [
        5      U " U" [        5      [        [
        5      [        [
        S-   5      [
        -  -   S[        4S4:X  d   eg )Nr   )MellinTransformf   T)	sympy.integrals.transformsrB   r   r   r;   r<   r#   r2   r   )rB   rC   s     ]/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/tests/test_transforms.pytest_undefined_functionrG   $   s    :AAaD!Q'?1Q4A+FFFFAaD3r7NAq1	1q!	$uQU|A~	52wEF F F    c                     [        S5      n [        U " [        5      [        [        5      R                  [        1:X  d   e[        U " [        5      [
        -  [        [        5      R                  [        [
        1:X  d   eg )NrC   )r   r   r;   r<   free_symbolsr=   )rC   s    rF   test_free_symbolsrK   ,   sW    AAaD!Q'44;;;AaDFAq)661a&@@@rH   c            
         SSK Jn   [        S5      n[        U" [        5      [        [
        5      R                  S5      U " [        [
        S-
  -  U" [        5      -  [        S[        45      :X  d   e[        U" [        5      [        [
        5      R                  S5      U " U" [        5      [        S[        -  [        -  [
        -  [        -  5      -  [        [        * [        45      :X  d   e[        U" [        5      [        [
        SS9R                  S5      U " U" [        5      [        [
        * [        -  5      -  [        S[        45      :X  d   e[        S	[        -  [        -  [        U" [
        5      [
        [        [        [         45      R                  S5      -  5      S
:X  d   e[        S	[        -  [        -  [#        U" [
        5      [
        [        5      R                  S5      -  5      S:X  d   e[%        U" [
        5      [
        [        5      R                  S5      U " U" [
        5      [        S	[        -  [        -  [
        -  [        -  5      -  [
        [        * [        45      :X  d   eg )Nr   )IntegralrC   rM   rD   Tnoconds   z.Integral(f(s)/x**s, (s, _c - oo*I, _c + oo*I))z2Integral(f(s)*exp(s*x), (s, _c - oo*I, _c + oo*I)))sympy.integrals.integralsrM   r   r   r;   r<   rewriter   r   r#   r   r   r   strr   r=   r>   r   r   )rM   rC   s     rF   test_as_integralrU   2   s   2AAaD!Q'//
;QUAaD1a*-. . .QqT1a(00<1c"Q$r'!)A+&&RC56 6 6QqT1a6>>zJ1c1"Q$i!Q,- - -qtAv.qtQAq6BJJ:VVW;< < <qtAv/!a;CCJOOP<= = =$QqT1a088D1c!A#b&(1*o%B3|45 5 5rH   c                  x   [        S5        [        n [        SSS9n[        [        [
        S-  -   5      [
        -   [        -  [        [        [
        S-  -   5      -  nU " UR                  [
        U* 5      [        [        5      S[        S-   -  S[        S[        -  -   -  -  U[        S[        -  -   S-
  -  -  [        [        [        -   5      -  [        S[        -
  S[        -  -
  5      -  [        S[        -
  5      -  [        [        5      * [        [        5      * S-  [        R                  -   4S4:X  d   e[        [        [
        S-  -   5      [
        -   [        -  nU " UR                  [
        U* 5      [        [        5      S[        S[        -  -   -  [        -  U[        S[        -  -   -  -  [        [        * S[        -  -
  5      -  [        [        [        -   5      -  [        [        * S-   5      -  [        [        5      * [        [        5      * S-  4S4:X  d   eU " UR                  [
        U* [        S05      [        [        5      US[        -  S-   -  * [        [        5      -  [        [        * [        R                  -
  5      -  S[        [        5      -  -  S[        SS5      4S4:X  d   eg )NzRisch takes forever.r>   TpositiverQ   rD   )r9   r   r   r&   r;   r>   r=   subsr<   r2   r!   r   Halfr   r   )MTbposexprs      rF   test_mellin_transform_failr_   E   sO    		 	B3&D QTNQ"4AqD>1DdiiD5!1a(
Aq1qs7|	#D1qs7Q;$7	7a!e	D
A!

"1q5\
*eVbeVAX&'	// / /
 QTNQ"DdiiD5!1a(AaCLN4!ac'?*5!a7 2 , AEl##(!a=1eVbeVAX	&& & & diiTE1a()1a0
1q/	%(	"5!aff#5	5qbz	B"a!4	)) ) )rH   c            
         SSK Jn Jn  [        n[	        SSS9nU" [
        [        -  [        [
        S-
  5      -  [
        [        5      S[        [        -   -  [        * [        [        5      * 4S4:X  d   eU" [
        [        -  [        S[
        -
  5      -  [
        [        5      S[        [        -   -  [        [        5      * [        4S4:X  d   eU" S[
        -
  [        S-
  -  [        S[
        -
  5      -  [
        [        5      [        [        5      [        [        5      -  [        [        [        -   5      -  S[        4[        [        5      S:  4:X  d   eU" [
        S-
  [        S-
  -  [        [
        S-
  5      -  [
        [        5      [        [        5      [        S[        -
  [        -
  5      -  [        S[        -
  5      -  [        * S[        [        5      -
  4[        [        5      S:  4:X  d   eU" S[
        -   [        * -  [
        [        5      [        [        5      [        [        [        -
  5      -  [        [        5      -  S[        [        5      4S4:X  d   eU" [        S[
        -
  5      [        * -  [
        [        5      S[        [         [        -  S-  5      -  [        S[        -
  5      -  [#        [         [        [        S-  -
  -  5      -  [        [        5      -  [        [        [        -
  5      -  [         -  S[        [        5      4[        [        5      S:  4:X  d   eU" S[
        -
  [        S-
  -  [        S[
        -
  5      -  [$        [
        S-
  [        S-
  -  -  [        [
        S-
  5      -  -   [
        [        5      nUS   (       d2   US   S[        [        5      * S-   4[        [        5      S:  4:H  5       eU" [
        [$        -  [&        [$        -  -
  [
        [&        -
  -  [
        [        5      S   [         [&        [$        [        -   S-
  -  -  [        [         [$        -  5      -  [        [         [        -  5      [        [         [$        [        -   -  5      -  -  :X  d   eU" [
        [$        -  U[$        -  -
  [
        U-
  -  [
        [        5      [         U[$        [        -   S-
  -  -  [        [         [$        -  5      -  [        [         [        -  5      [        [         [$        [        -   -  5      -  -  U " S[        [$        5      * 5      U" SS[        [$        5      -
  5      4S4:X  d   e[)        [
        [&        S-  -   5      [&        -   [$        -  nU" UR+                  [&        U5      [
        [        5      [$        * SU-  [$        S[        -  -   -  -  [        [        5      -  [        [$        * S[        -  -
  5      -  [        [$        * [        -
  S-   5      -  S[        [$        5      * S-  4S4:X  d   e[)        [
        [&        S-  -   5      [&        -   [$        -  [)        [
        [&        S-  -   5      -  nU" UR+                  [&        U5      [
        [        5      S[$        S[        -  -   -  U[$        S[        -  -   S-
  -  -  [        [        5      -  [        S[$        -
  S[        -  -
  5      -  [        S[$        -
  [        -
  5      -  S[        [$        5      * S-  [,        R.                  -   4S4:X  d   eU" [1        [
        * 5      [
        [        5      [        [        5      S[        4S4:X  d   eU" [1        S[
        -  5      [
        [        5      [        [        * 5      [        * S4S4:X  d   eU" [3        [
        5      S	-  [        S[
        -
  5      -  [
        [        5      S
[        S-  -  S[        4S4:X  d   eU" [3        [
        5      S-  [        [
        S-
  5      -  [
        [        5      S[        S	-  -  [        * S4S4:X  d   eU" [3        [
        S-   5      [
        [        5      [         [        [        [         [        -  5      -  -  SS4:X  d   eU" [3        S[
        -  S-   5      [
        [        5      [         [        [        [         [        -  5      -  -  SS4:X  d   eU" [3        [        S[
        -
  5      5      [
        [        5      [         [        [5        [         [        -  5      -  -  SS4:X  d   eU" [3        [        SS[
        -  -
  5      5      [
        [        5      [         [        [5        [         [        -  5      -  -  SS4:X  d   eU" [7        [)        [
        5      5      [
        [        5      [        [        [,        R.                  -   5      * [)        [         5      [        -  -  [9        SS5      S4S4:X  d   eg )Nr   MaxMinr>   TrW   rD   rY   rQ                  rY   r   r   rD   )(sympy.functions.elementary.miscellaneousrb   rc   r   r   r;   nur/   r<   r   r!   betar2   rhoabsr)   r   r(   r=   r>   r&   rZ   r   r[   r#   r%   r*   r0   r   )rb   rc   r\   r]   mtr^   s         rF   test_mellin_transformrq   c   s   C	B3&D aeIa!e$$a+	R!VsRVGnd+, , ,aeIa!e$$a+	
BFr"vgr]D)* * * q1uq!)AE"22Aq9	tU1X	eD1Ho	-2w41EF F Fq1uq!)AE"22Aq9	tU1t8a<(	(q1u	5S!bh,DA	// / / q1uoq!$	q%a.	 s	+aC\4@A A A c!a%jC4 !Q'	#bfQh-a#g&BCE	N	!!H	%%*3q5\	224	5	
BsGbgk,# # # # 
QUdQh	!a% 00Q$(##Ia!e$44567
<Ba5="Q%QD	A.41===5q!tad{QU#Q*1-
1q1uqy>#bd)#SAYs2q1u:%>?@ @ @q!tdAg~D)1a0	D1q519	c"Q$i	'RT3r1q5z?)B	CRUF^SA1I./	77 7 7 QTNQ"Ddii4 !Q'
QtVq1Q3w	a	(rAaCx	8rAvz9J	JAqM4	!! ! ! QTNQ"4AqD>1Ddii4 !Q'	
Q1WdQ1Wq[)	)%(	2*/A!*<
==B1q519=M
NAq166!"D	** * * c1"gq!qAr7D 9999c"Q$iA5!9sAh"==== c!fai	!a%((!Q/Bq!tGaWd3KKKKc!fai	!a%((!Q/AadFbS!Hd3KKKKc!a%j!QB#bd)$4gt#DDDDc!A#'lAq!b!C1I+&6%EEEEc#a!e*oq!$Qs2a4y[)97D(IIIIc#a!A#g,A&2qRT{+;VT*JJJJ c$q'lAq!
AFF

	T"XaZ	(8B?A*>EF F FrH   c                  >   [         n U " [        [        5      [        S-   -  [        [        5      nUSS  S:X  d   e[	        US   SS9R                  [        5      (       a   eU " [        [        5      S-  [        S-   -  [        [        5      nUSS  S:X  d   e[	        US   SS9R                  [        5      (       a   eU " [        [        5      [        S-   S-  -  [        [        5      nUSS  S:X  d   e[	        US   SS9R                  [        5      (       a   eg )NrD   )rj   Tr   T)allow_hyperrQ   ))r   rQ   T)r   r%   r;   r<   r5   hasr3   )r\   rp   s     rF   test_mellin_transform2ru      s   	B 
CFAENAq	!Bab6^###2a5d377@@@@	CFAIq1uq!	$Bab6^###2a5d377@@@@	CFAEA:q!	$Bab6^###2a5d377@@@@@rH   c                  %   SSK Jn   [        nU" [        [        S[        [        5      -  5      [        [        5      [        [        S-  [        -   5      [        [        S-  [        -
  S-   5      -  [        [        5      * S-  [        SS5      4S4:X  d   eU" [        [        [        5      5      [        [        [        [        5      5      -  [        [        5      S[        -  [        S[        -  [        R                  -   5      -  [        [        S-  [        -   [        R                  -   5      -  [        [        * S-  [        -
  S-   5      [        [        S[        -  -
  S-   5      -  -  [        [        5      * S-  [        R                  -
  [        SS5      4S4:X  d   eU" [        [        [        5      5      [        [        [        [        5      5      -  [        [        5      S[        -  [        [        S-  [        -   5      -  [        S[        -  [        R                  -   5      -  [        [        * S-  [        -
  [        R                  -   5      [        [        S[        -  -
  S-   5      -  -  [        [        5      * S-  [        SS5      4S4:X  d   eU" [        [        [        [        5      5      S-  [        [        5      [        [        [        -   5      [        [        R                  [        -
  5      -  [        [        5      [        S[        -
  5      -  [        S[        -   [        -
  5      -  -  [        [        5      * [        R                  4S4:X  d   eU" [        [        [        [        5      5      [        [        * [        [        5      5      -  [        [        5      [        [        5      [        [        R                  [        -
  5      -  [        [        5      [        S[        -
  [        -
  5      -  [        S[        -   [        -
  5      -  -  S[        R                  4S4:X  d   eU" [        [        S-
  [        [        5      5      [        [        [        [        5      5      -  [        [        5      [        S[        -
  5      [        [        [        -   [        R                  -
  5      -  [        [        5      [        [        SS5      [        -
  5      -  [        [        [        -
  [        R                  -   5      -  -  [        R                  [        [        5      -
  [        R                  4S4:X  d   eU" [        [        [        [        5      5      [        [         [        [        5      5      -  [        [        5      S[        -  [        SS[        -  -
  5      -  [        [        [         -   S-  [        -   5      -  [        S[        -
  [         [        -
  S-  -   5      [        S[        -
  [        [         -
  S-  -   5      -  [        S[        -
  [        [         -   S-  -   5      -  -  [        [        5      [        [         5      -   * S-  [        R                  4S4:X  d   eU" [        [        [        [        5      5      S-  [        [        * [        [        5      5      S-  -   [        [        5      SS  U " [        [        5      [        [        5      * 5      [        R                  4S4:X  d   eU" [#        [        S[        [        5      -  5      [        [        5      [        [        [        S-  [        -
  -  5      * [        [        [        S-  -
  5      -  [        [        [        S-  -   5      -  [        -  U " [        [        5      * S-  [        [        5      S-  5      [        SS5      4S4:X  d   eU" [        [        [        5      5      [#        [        [        [        5      5      -  [        [        5      S[        -  * [        [        [        S-  [        -
  -  5      -  [        [        R                  S[        -  -
  5      -  [        S[        -
  S-  [        -   5      -  [        S[        -   S-  [        -   5      -  [        [        5      [        S[        -
  [        S-  -
  5      -  [        S[        -
  [        S-  -   5      -  -  U " [        [        5      S-   * S-  [        [        5      S-
  S-  5      [        SS5      4S4:X  d   eU" [        [        [        5      5      [#        [        [        [        5      5      -  [        [        5      S[        -  * [        [        [        S-  [        -
  -  5      -  [        [        [        S-  -
  5      -  [        [        [        S-  -   5      -  [        [        R                  S[        -  -
  5      -  [        [        5      [        [        R                  [        -
  [        S-  -
  5      -  [        [        R                  [        -
  [        S-  -   5      -  -  U " [        [        5      * S-  [        [        5      S-  5      [        SS5      4S4:X  d   eU" [        [        [        [        5      5      [#        [        [        [        5      5      -  [        [        5      [        [        [        -  5      * [        [        5      -  [        [        [        -   5      -  [        [        R                  [        -
  5      -  [        [        S	5      -  [        S[        -   [        -
  5      -  -  U " [        [        5      * S5      [        R                  4S4:X  d   eU" [        [        [        [        5      5      [#        [         [        [        5      5      -  [        [        5      S[        -  * [        [        [        S-  [         S-  -
  [        -   -  5      -  [        SS[        -  -
  5      -  [        [        S-  [         S-  -
  [        -   5      -  [        [        S-  [         S-  -   [        -   5      -  [        [        [        S-  [         S-  -
  [        -
  S-   5      -  [        [        S-  [         S-  -   [        -
  S-   5      -  -  U " [        [        5      * [        [         5      -   S-  [        [        5      * [        [         5      -
  S-  5      [        R                  4S4:X  d   eU" [#        [        [        [        5      5      S-  [        [        5      SS  U " [        [        5      * S[        [        5      5      [        R                  4S4:X  d   eU" [%        [        S[        [        5      -  5      [        [        5      [        [        [        S-  -
  5      [        [        [        S-  -   5      -  S-  U " [        [        5      * S-  [        [        5      S-  5      [&        4S4:X  d   eU" [        [        S[        S[        [        5      -  5      -  5      [%        [        S[        S[        [        5      -  5      -  5      -  [        [        5      S[        * -  [        S[        -  5      -  [        [        S-  [        -   5      -  S[        [        S-  [        -
  S-   5      -  -  U " S[        [        5      * S-  5      [&        4S4:X  d   eU" [)        [        [        [        5      5      [%        [        [        [        5      5      -  [        [        5      [        [        5      [        [        [        -   5      -  [        [        * [        R                  -   5      -  S[        [        5      -  [        [        [        -
  S-   5      -  -  U " [        [        5      * S5      [        R                  4S4:X  d   eU" [)        [         [        [        5      5      [%        [        [        [        5      5      -  [        [        5      SS[        -  S-
  -  [        S[        -  S-   5      -  [        [        * S-  [         S-  -   [        -   5      -  [        [        S-  [         S-  -   [        -   5      -  [        [        * S-  [         S-  -   [        -
  S-   5      [        [        S-  [         S-  -   [        -
  S-   5      -  -  U " [        [        5      * S-  [        [         5      S-  -
  [        [        5      S-  [        [         5      S-  -
  5      [        R                  4S4:X  d   eU" [+        [        * S-  5      [%        [        [        S-  5      -  [        [        5      n[-        [/        [-        US   R1                  SS
95      5      5      nUS[        [        SS5      -  -  [        [        [        -  5      -  [        [        R                  [        -
  5      -  [        S[        -  [        -  5      [        S[        -  [        -  5      -
  [        [        * [        -
  S-   5      -  [        [        [        -
  S-   5      -  -  :X  d   eUSS  U " [        [        5      * [        [        5      5      [&        4S4:X  d   eg )Nr   rb   rQ   rD   rg   rd   TrN   z3/2func)rk   rb   r   r,   r=   r&   r;   r<   r2   r!   r   r)   r   r[   r(   r   r>   r.   r-   r   r+   r#   r4   r6   expand)rb   r\   rp   mt0s       rF   test_mellin_transform_besselr|      s3   <	B ga47#Q*	qsQwacAgk*	*beVAXx1~,FMN N Nc$q'l71d1g..15	
AeBqD166M"	"51q166)9#:	:qbdQhlE!ac'A+..
0	Aq1668Aq>2+,0	22 2 2 c$q'l71d1g..15	
AeAaC!Gn	U2a4!&&=1	1qbdQh q1Q3w{!33
5	Aq(1a.7"#'	)) ) ) gaa!1$a+	q1ueAFFQJ'	'HU1q5\!%A	"22
4eVQVVd	$$ $ $ gaa!'1"d1g"661=	q%
#	#HU1q519%%eAEAI&66
8K	   ga!eT!W%gaa&991a@	q1ueAEAFFN+	+HU8Aq>A-..uQUQVV^/DD
FVVbe^QVV$d	,, , , gaa!'!T!W"55q!<	
AeA!Gn	UAE19q=1	1!a%1q5!)#$U1q5AE19+<%==AEQUAI%&'
( !ur!u}a($	00 0 0
 gaa!1$wr47';Q'>>1EabI
bebeV
aff	%t,- - - ga47#Q*
b!A#'l
	E!ac'N	*5QqS>	9"	<"Q%2a57#Xa^4d	<< < < c$q'l71d1g..15
Q$s2qsQw< 	 qvv!|!4	4!a%Q
  %q1uai!m 4
5HU1q51Q3;''a!eack(::
< 2a519a"Q%!)Q/!Q@$	HH H H
 c$q'l71d1g..15
Q$s2qsQw< 	 q1Q3w	/a!A#g	>uQVVaPQc\?R	RHU166A:!+,,U166A:!3C-DD
F"Q%2a57#Xa^4d	<< < < gaa!'!T!W"55q!<
bd)E!H	U1q5\	)%
*;	;%Lq1uqy))
+"Q%^QVV$d	,, , , gaa!'!T!W"55q!<
Q$s2qsQqSy1}%&	&uQ1W~	51qsQ
  %acAaCi!m 4
5uQqS1Q3Y]Q&''acAaCi!ma.?(@@
B 2a5&2a5.!#r!ufr!una%78!&&A4	II I I gaa!1$a+AB/BqE61be$aff-t45 5 5 ga47#Q*		
QqS
AaC.
!!"
#%("Q%2a57%;R$@$	HH H H ga4$q'	?*+G	1T!DG)_-  !%)*aRqs);acAg*%!a!,,*.03A1vax0@"/Et)MM M M gaa!'!T!W"55q!<	q%	A 
aR!&&[!
"#$T"X:eAEAI.>#>
@	beVQ $	(( ( ( gaa!'!T!W"55q!<	
QqS1WeBqD1Ho	%eQBqD1Q3JN&;	;acAaCi!m
#QBqD1Q3JNQ$67acAaCi!ma !
"$'Aq2a57(:
1a"Q%'%FF$$%)	++ + + 
C1Iga1o%q!	,B
HYr!u|||'>?@
AC!BA&&s2a4y0qvvz1BB	QrT!Vs1R46{	"E1"q&1*$55eAEAI6FFH H H Hab6sBqE62a5)2.5555rH   c                     SSK Jn   SSKJnJnJn  SSKJn  [        SSS9n[        SSS	9n[        U" [        5      [        [        5      [        [        5      [        -  S[        4S4:X  d   e[        [        [        5      [        -  [        [        S[        45      R                  [         5      R#                  5       U" [        5      :X  d   e[        [!        [$        [        5      [        [        5      [        [        5      [$        [        -   S
-
  -  U " S
['        [$        5      -
  S5      [        4S4:X  d   eU" [)        [        [        [        5      U[        -   S
-
  -  [        [        S
U-
  [        45      R                  [         5      R#                  SS95      5      [!        U[        5      :X  d   e[        U" [        5      [        [        5      S[        -  * [+        [,        5      -  [        [        S-  [.        R0                  -   5      -  S[        -  [        [        * S-  S
-   5      -  -  SS4:X  d   e[        S[        -  * [+        [,        5      -  [        [        S
-   S-  5      -  S[        -  [        [        * S-  S
-   5      -  -  [        [        S5      U" [        5      :X  d   e[        U" [+        [        5      5      [        [        5      SS[        -  S
-
  -  * [+        [,        5      -  [        [        5      -  [        [        [        * [.        R0                  -   5      -  -  SS4:X  d   e[        S[        -  * [+        [,        5      -  [        [        5      -  S[        -  [        [        * [.        R0                  -   5      -  -  [        US5      R#                  5       U" [+        U5      5      :X  d   eg )Nr   rw   )CiE1Sisimplifyr=   Tnegativeu)polarrD   rx   rQ   ri   rj   rd   )rk   rb   'sympy.functions.special.error_functionsr~   r   r   sympy.simplify.simplifyr   r   r   r;   r<   r2   r   r   rS   r1   rz   r=   r!   r"   r&   r   r   r[   )rb   r~   r   r   r   anegr   s          rF   test_expintr     s   <BB0#%Ds$ABqE1a(U1XaZ!R$,GGGG#E!HQJ1"gwvvvx2a59 9 9F1aL!Q/	q1q519	A1Iq 126=> > > J,U1Xtax!|-Dat8R."")'&/&&d&2CE F 	tQ  
 BqE1a(
Q$tBxacAFFl+	+	!E1"Q$(O
%t	-- - - $QTE$r(N5!a%3C$C&'c%1q/&9%;<=q'Ka5   BtAwKA.
acAg,tBx	a	(!E1"qvv+,>*>	?NO O O#	
Ad2huQx1UA2;%7!78	1ffh"T!W+. . .rH   c                  (+  ^^ SSK Jn   SSKJnJn  SSKJn  SSKJm  SSK	J
m  [        nU" [        [        5      [        [        S[        45      [!        [        * 5      :X  d   eU" [        [        * 5      [        [        [        * S45      [!        S[        -  5      :X  d   eT" U" [        S[        S-  -  S-
  -  [        [        S[        45      5      [        S-  S	-   [#        S	[        -
  5      -  S
[        -  -  :X  d   eU" S	[        S-  S	-
  -  [        [        S5      [        * [#        [        * S	-   5      -  S-  [#        [        S	-
  5      S[        -  -  -
  :X  d   eU" S	[        S-  S	-
  -  [        [        S5      [        * [#        [        * S	-   5      -  S-  [#        [        S	-
  5      S[        -  -  -
  :X  d   eU" [        [        5      [        [        S	-
  5      -   [        [        S	[        45      [        S	-   [!        [        * 5      -  [        -  :X  d   e[%        SSS9nU" S	[        S-  S	-   -  [        [!        [        * 5      S [        45      R'                  [        U5      R)                  [*        5      R                  5       [+        U5      [#        S	[!        U* 5      -
  5      -  :X  d   e[%        SSS9u  pgU" U[        * U-  -  [-        [        U-  5      -  [        -  [        [        S[        45      [!        U* [        U-  -  5      :X  d   eU" [-        Xg-  [        U-  -   5      U[        -   -  [        [        U* [        45      [        U-  [!        [        U-  * 5      -  :X  d   eUU4S jn[%        SSS9n	U" SU	[        -   -  [        [        [        * S 45      [        U	-  [#        [        S	-
  5      -  :X  d   eU" S	U	[        -   -  [        [        S [        45      [        U	-  [#        S	[        -
  5      -  :X  d   eU" U" [        [.        5      [        [        5      -  [        [        [.        -   5      -  [        [        S[        45      5      S	[        -
  [.        S	-
  -  [#        S	[        -
  5      -  :X  d   eU" U" [        [.        5      [        S	[.        -
  [        -
  5      -  [        S	[        -
  5      -  [        [        [        * S 45      5      [        S	-
  [.        S	-
  -  [#        [        S	-
  5      -  :X  d   eU" U" [        [        5      [        [0        [        -
  5      -  [        [0        5      -  [        [        S5      5      S	[        S	-   -  [0        -  :X  d   eU" [2        [4        -  [2        [        S	-
  -  -  [+        [6        [4        -  5      -  [        [        5      -  [        [        [4        -   5      -  [        S	[        -
  5      -  [        S	[        -
  [4        -
  5      -  [6        -  [        [        U" [9        [4        5      * S5      U" S	[9        [4        5      -
  S	5      45      n
T" [;        U
SS95      R=                  [>        [         5      R                  5       [2        [4        -  * [        [4        -  -   [2        * [        -   -  :X  d   eT" U" S	[A        [6        5      -  [4        * S-  -  [        [        5      -  [        S	[4        -
  S-  [        -
  5      -  [        [4        * S-  [        -
  5      -  [        S	[4        -
  [        -
  5      -  [        [        S[9        [4        5      * S-  45      5      S	[A        [        S	-   5      -   [4        -  :X  d   eT" U" S[B        S[        -  -   -  [D        [B        S[        -  -   S	-
  -  -  [        [        5      -  [        S	[B        -
  S[        -  -
  5      -  [        S	[B        -
  [        -
  5      -  [        [        S[9        [B        5      * S	-   S-  45      5      [D        [B        S	-
  -  [D        S-  [A        S	[        [D        S-  -  -   5      S	-   [B        -  -  [        [A        S	[        [D        S-  -  -   5      S	-   [B        S	-
  -  -  -   -  [D        S-  [        -   -  :X  d   eT" U" S[4        S[        -  -   -  * [4        -  [D        [4        S[        -  -   -  -  [        [        5      -  [        [4        * S[        -  -
  5      -  [        [4        * [        -
  S	-   5      -  [        [        S[9        [4        5      * S-  45      5      [D        [4        -  [A        S	[        [D        S-  -  -   5      S	-   [4        -  -  :X  d   eU" S[        S-  -  [        [        S[        45      [G        [        5      S
-  [#        S	[        -
  5      -  :X  d   eU " U" S[        S
-  -  [        [        [        * S45      SS9[G        [        5      S-  [#        [        S	-
  5      -  :X  d   eU" [6        [        [+        [6        [        -  5      -  -  [        [        S5      [G        [        S	-   5      :X  d   eU" [6        [        [+        [6        [        -  S-  5      -  -  [        [        S5      [G        [        S-  S	-   5      :X  d   eU" [6        [        [+        S[6        -  [        -  5      -  -  [        [        [I        SS5      S45      [G        [A        [        5      S	-   5      :X  d   eU" [6        [        [+        [6        [        -  5      -  -  [        [        S5      [G        S	S	[        -  -   5      :X  d   eS nU" U" U" [6        [        [K        [6        [        -  5      -  -  [        [        S5      5      5      [G        S	[        -
  5      [#        S	[        -
  5      -  [G        [        S	-
  5      [#        [        S	-
  5      -  -   [G        [        5      [#        [        S	-
  5      -  [G        S	S	[        -  -
  5      [#        [        S	-
  5      -  -   [G        [        * S	-   5      [#        [        * S	-   5      -  -   4;   d   eU" U" [6        U" [6        [        -  5      -  [        -  [        [        S5      5      [G        S	[        -  S	-
  5      [#        S	[        -
  5      -  [G        S	S	[        -  -
  5      [#        [        S	-
  5      -  -   [G        [        5      * [#        [        * S	-   5      -  [G        S	S	[        -  -
  5      [#        [        S	-
  5      -  -   [G        [        * S	-   5      [#        [        * S	-   5      -  -   4;   d   eU" [        [        [L        RN                  -   5      * [A        [6        5      [        -  -  [        [        [I        SS5      S45      [Q        [A        [        5      5      :X  d   eT" U" [        [B        S-  [        -   5      [        [B        S-  [        -
  S	-   5      -  [        [        [9        [B        5      * S-  [I        SS
5      45      5      [S        [B        S[A        [        5      -  5      :X  d   eT" U" S[B        -  [        [L        RN                  S[        -  -
  5      -  [        [        [B        S	-   S-  -   5      -  [        S	[        -
  [B        S-  -
  5      [        S	S[        -  -
  [B        -   5      -  -  [        [        [9        [B        5      S	-   * S-  [I        S	S
5      45      5      [+        [A        [        5      5      [S        [B        [A        [        5      5      -  :X  d   eT" U" S[B        -  [        [B        S-  [        -   5      -  [        [L        RN                  S[        -  -
  5      -  [        [L        RN                  [        -
  [B        S-  -
  5      [        S	S[        -  -
  [B        -   5      -  -  [        [        [9        [B        5      * S-  [I        S	S
5      45      5      [U        [A        [        5      5      [S        [B        [A        [        5      5      -  :X  d   eT" U" [        [B        [        -   5      [        [L        RN                  [        -
  5      -  [A        [6        5      [        S	[        -
  5      -  [        S	[B        -   [        -
  5      -  -  [        [        [9        [B        5      * [L        RN                  45      5      [S        [B        [A        [        5      5      S-  :X  d   eT" U" [        [        5      [        [L        RN                  [        -
  5      -  [A        [6        5      [        S	[        -
  [B        -
  5      -  [        S	[B        -   [        -
  5      -  -  [        [        S[L        RN                  45      5      [S        [B        * [A        [        5      5      [S        [B        [A        [        5      5      -  :X  d   eT" U" S
[        -  [        S [        -  S	-   5      -  [        [B        S-  [D        S-  -   [        -   5      -  [        [B        * S-  [D        S-  -   [        -
  S	-   5      [        [B        S-  [D        S-  -
  [        -
  S	-   5      -  [        [B        S-  [D        S-  -   [        -
  S	-   5      -  -  [        [        [9        [B        5      [9        [D        5      -   * S-  [L        RN                  45      5      [S        [B        [A        [        5      5      [S        [D        [A        [        5      5      -  :X  d   eT" U" SS[        -  -  * [U        [6        [B        -  S-  [6        [D        -  S-  -
  [6        [        -  -   5      -  [        S [        -  S	-   5      -  [        [B        S-  [D        S-  -
  [        -   5      -  [        [B        S-  [D        S-  -   [        -   5      -  [6        [        [B        S-  [D        S-  -
  [        -
  S	-   5      -  [        [B        S-  [D        S-  -   [        -
  S	-   5      -  -  [        [        U" [9        [B        5      * S-  [9        [D        5      S-  -
  [9        [B        5      * S-  [9        [D        5      S-  -   5      [L        RN                  45      5      [S        [B        [A        [        5      5      [S        [D        * [A        [        5      5      [S        [D        [A        [        5      5      [U        [6        [D        -  5      -  -
  * -  [+        [6        [D        -  5      -  :X  d   eU" [6        [U        [6        [        -  5      -  [        [        S[L        RN                  45      [A        [        5      [        S	-   -  :X  d   eg )!Nr   rz   ra   )cotpowsimpr   rY   rQ   rD   rd   )rY   NNrD   rTrealza brW   c           	      ^   > T" T" [        U SS9SS95      R                  [        [        5      $ )NFdeepTforce)r   replacer$   r#   )r^   r   r   s    rF   	simp_pows0test_inverse_mellin_transform.<locals>.simp_powsW  s+    
4e <DIJRRS\^abbrH   rl   )r   NFr   re   rf   rh   r   rg   ri   )rN   r   rj   c           	      z    SSK Jn  SSKJn  SSKJn  U" U" U" U SS9SSS9SS9R                  [        [        5      $ )Nr   r   r   )
logcombineTr   )r   r   )	sympy.core.functionrz   sympy.simplify.powsimpr   r   r   r   r$   r#   )r^   rz   r   r   s       rF   mysimp-test_inverse_mellin_transform.<locals>.mysimp  s:    .26Jt404H	3/	0rH   rN   )+r   rz   rk   rb   rc   (sympy.functions.elementary.trigonometricr   r   r   r   r   r   r2   r<   r;   r   r#   r/   r   rZ   rS   r)   r    rm   rn   r@   r?   r   r!   r   r   r$   r&   r=   r>   r%   r   r*   r   r[   r0   r,   r(   )rz   rb   rc   r   IMTr   _a_br   rl   r^   r   r   r   s               @@rF   test_inverse_mellin_transformr   5  s   *C<.0
"CuQxA2w'3r7222uaRy!Q"a)SAY666C1QT6A:1q"g67	
A9QU##QqS)* * * q!Q$(|Q:.	
9aR!VQ1q5!11Q3!778 8 8q!Q$(|Q9-	
9aR!VQ1q5!11Q3!778 8 8 uQx%A,&1q"g61q5#qb'/!:KKKK 	$Aq!Q$(|QQB$ Q
773<
q6)AQBK(() ) )
 UT*FBrQBrE{9QrT?*1,aQG<RC2INNNy2&Q/ARyAQU3PQSUPUv;EVVVVc 
D	!Br26{AqB3+.!R%	!a%8H2HHHHq"q&z1a$,2iA6F0FFFFStU1X-eAHo=q!aWMNETAXyQ//0 0 0StU1t8a<%88q1uEAT{, -ETAXyQ//0 0 0 Sq%a.0s;Q9MNq1uI  q!tAAJs2a4y((##(Q<005a!e==B1q519=MNNPQARUFAA1Iq0ABDD :d/088CHQQSTEAqDLA26"# # # C$r(
QBqD)%(25!a%Q3GGr!tax)).q1uqy)9:1q2a5&(m- . 
T!a%[1   CA!GQQqS1%55eAh>uQUQqS[?QQq1uqy)*+,a!r!ufqj!^1DF G	AE
AqD$q1QT6z*Q.22QQ1a4Z8H18LE8 6  	TAX	   CQ1WaA!G4U1X=eQB1HoMQ
+,-.A1vax=B C	1d1qAv:"Q&&' ' '
 r!Q$w1q"g&#a&!)Ia!e4D*DDDD#a1fabS!H-T:A	)AE""# # #r1SAY;Aw/3q1u:===r1SAa[=!1a1SA]BBBr1S2a[=!1a(2q/1)=>#d1gPQkBRRRRr1SAY;Av.#a!A#g,>>>0 &R3r!t9-q!W=>?AE
9QU##c!a%j1q51A&AAAyQ#a!A#g,yQ/?"??#qb	G
 CaR!VC 	D   
 #bRTl1naF34AaC!GYq1u%%A!GYq1u5E(EE	Q	1"q&!!CAaCL1q51A$AAC	I
 EaR!VE 	9        a!&&j!!48A:.1xA6JKDG   CacAguQqS1Wq['991a2a5&(HUVXYNA[\]1aQi ! ! !C1U166AaC<00qAE19}1EEq1uqs{+E!ac'A+,>>@r!uqy\!^Xa^<> ? 	DGWQQ(() ) ) C1U1Q37^+E!&&1Q3,,??qvvzAaC/0q1Q3w{1CCEbeVAXx1~68 9 	DGWQQ(() ) )
 Ca!eU166A:%66b%A,.uQUQY/??AbeVQVV,. / 	47Q   Caqvvz!22b%A	"225Q3CCEQK) * 	DGWQQ001 1 1 C1U2a4!8_,U1Q319q=-AAr!taczA~1251qsQ9J3KK!ac	A 1234 r!ur!u}-a/8: ; 	47GAtAw//	0 0 0 CQqS	#bd1fr!tAvo1&<"==eBqD1HoM!A#!)a-()).qsQqSy1})=>acAaCi!ma/00qsQqSy1}q7H1IIK q"Q%BqE!G+beVAX1a-?@!&&I	K L
 AtAw''1"d1g*>AtAw'BqD	1+2 )3 336r!t9== = = r#bd)|QAqvv;/47AE?BBBrH   c                  P  ^
^^^ SSK Jm
JmJm  SSKJn   SSKJm  [        n[        nU
UUU4S jnS n[        SSS	9n[        S
5      n[        SSS9n[        SSS9n[        SSS9n	[        U" [        5      [        U5      [        U" [        5      [        U5      :X  d   e[        U" U5      U[        5      [        U" U5      U[        5      :X  d   eU" U" [        S[!        SU-  [        -  5      -
  5      [        U5      5      U" XW-  5      U-  :X  d   eU" U" [        S[!        U[        -  5      -
  5      S[!        U[        -  5      -
  -  [        U5      5      U" XW-  5      S-  U-  :X  d   eU " U" [#        U* [        -  5      [        [        5      -  [        U5      [$        S9SUS[&        -  [$        -  U-  -   -  :X  d   eU" SUS[&        -  [$        -  [        -  -   -  [        U	SS9[#        U* U	-  5      S4:X  d   eU" SUS[&        -  [$        -  [        -  -   -  [        U	* SS9S:X  d   eU" SUS[&        -  [$        -  [        -  -   -  [        [        SSS9SS9S:X  d   eU " U" [        [#        U* [        -  5      -  [        [        5      -  [        U5      [$        S9SUS[&        -  [$        -  U-  -   S-  -  :X  d   eU" [#        U* [        -  5      [)        U[        -  5      -  [        [        5      -  [        U5      XS-  US[$        -  [&        -  U-  -   S-  -   -  :X  d   eU" [#        U* [        S-  -  5      [        U5      [+        [&        5      [#        [&        S-  * US-  -  U-  5      -  [+        U5      -  :X  d   eU" [+        [&        U-  5      [#        [&        U-  S-  * U-  5      -  U[        5      [#        U* [        S-  -  5      :X  d   eU" [#        U* [!        [        5      -  5      [        U5      SU-  US-  S[&        S-  -  US-  -  -   -  :X  d   eg )Nr   )rz   expand_complexexpand_trig)factorr   c           	      8   > T" T" T" T" U 5      5      5      5      $ N )r;   rz   r   r   r   s    rF   simp$test_fourier_transform.<locals>.simp  s    N6!9$=>??rH   c                 :    [        [        U -  5      [        U -  -  $ r   )r)   r   )r;   s    rF   sinc$test_fourier_transform.<locals>.sinc  s    2a4y"Q$rH   kTr   rC   r=   rW   r>   poskrD   rQ   )	extensionFrO   )r   Tr   rd   )r   rz   r   r   sympy.polys.polytoolsr   r   r   r   r   r   r   r;   r   r   r/   ro   r#   r   r   r)   r&   )r   FTIFTr   r   r   rC   r=   r>   r   rz   r   r   r   s             @@@@rF   test_fourier_transformr     s   II,0	B
#C@ @ $AA 	d#Ad#A6D)D QqT1a(,<QqT1a,HHHH$	!a.qtQ:; ; ; 9QQqSU^,a34QS	!CCC9QQqS\*AAaCL91a@AT!#YPQ\RS^SSS "S!AYy|+Q2a@	1qtAvax<   q!ad1fQh,D"A2d7|T23 3 3q!ad1fQh,TE'( ( (q!ad1fQh,GC$$?'( ( ( "QsA2a4y[1-q!4B	1qtAvax<!
  c1"Q$iAaC 1-q!4	a41qs2vax<!##$% % % c1"QT'lAq!T"Xc2q5&A+a-.@%@a%HHHHtBqDz#1qyjl++Q2c1"QT'lBBBc1"SV)na#qsAqD1RU71a4<,?'@@@@rH   c            	         [        S5      n [        S5      n[        S5      n[        S5      n[        U" U 5      X5      [        U" U 5      X5      :X  d   e[	        U" U5      X5      [        U" U5      X5      :X  d   e[        S[        U 5      -  X5      S[        U5      -  :X  d   e[	        S[        U5      -  X5      S[        U 5      -  :X  d   e[        S[        U 5      -  S-  X5      S[        U5      -  :X  d   e[        X* -  X5      SU* [        R                  -   -  XS-
  -  -  [        U* S-  S-   5      -  [        US-   S-  5      -  :X  d   e[	        SU* [        S5      S-  -   -  XS-
  -  -  [        U* S-  S-   5      -  [        US-  [        R                  -   5      -  X5      X* -  :X  d   e[        [        U* U -  5      X5      [        S5      U-  [        [        5      US-  US-  -   -  -  :X  d   e[	        [        S5      U-  [        [        5      US-  US-  -   -  -  X5      [        U* U -  5      :X  d   e[        [        U 5      U -  X5      [        S5      [        [        5      -  [        US-  5      S[        -  -   * -  S-  :X  d   e[        U [        U* U S-  -  5      -  X5      [        S5      U-  [        US-  * SU-  -  5      -  SU[        SS5      -  -  -  :X  d   e[	        [        S5      U-  [        US-  * SU-  -  5      -  SU[        SS5      -  -  -  X5      U [        U* U S-  -  5      -  :X  d   eg )	Ntwr=   rC   rD   rg   rQ   rd   )r   r   r   r   r   r   r&   r   r[   r2   r#   r   r%   r   r   )r   r   r=   rC   s       rF   test_sine_transformr     s   AAAA !A$%qtQ)BBBB!	!a+AaD!78 8 8 !DG)Q*aQi777!!DG)Q2aQi???1T!W9q.!/1T!W9<<<!b'1(A	
QVV-QZ-  %qbdQh-005q1uai0@-A A A A!!qb1	,, ' #a%j#r!tax#)).qsQVV|)<#=>?DGH2wO O O QBqD	1 GAItBxA1'=>? ? ?!Q	48QTAqD[)*A258!AY? ? ? Aq!748+c!Q$i!J,.F,GGIJ J J 	#qbAg,!%a31uac{+;!;Qq(1a.?P=P!QR R R!Q	#q!teQqSk""Aa!Q&7$78!@CDS!AqD\>R R RrH   c            
      |	   SSK Jn Jn  [        S5      n[        S5      n[        S5      n[	        S5      n[        U" U5      X#5      [        U" U5      X#5      :X  d   e[        U" U5      X25      [        U" U5      X25      :X  d   e[        S[        U5      -  X#5      S[        U5      -  :X  d   e[        S[        U5      -  X25      S[        U5      -  :X  d   e[        SUS-  US-  -   -  X#5      [        S5      [        [        5      -  [        U* U-  5      -  SU-  -  :X  d   e[        UU* -  X#5      SU* [        R                  -   -  X4S-
  -  -  [        U* S-   S-  5      -  [        US-  5      -  :X  d   e[        SU* [        S5      S-  -   -  X4S-
  -  -  [        U* S-  [        R                  -   5      -  [        US-  5      -  X25      X$* -  :X  d   e[        [        U* U-  5      X#5      [        S5      U-  [        [        5      US-  US-  -   -  -  :X  d   e[        [        S5      U-  [        [        5      US-  US-  -   -  -  X25      [        U* U-  5      :X  d   e[        [        U* [        U5      -  5      [        U[        U5      -  5      -  X#5      U[        US-  * SU-  -  5      -  SU[!        S	S5      -  -  -  :X  d   e[        SXB-   -  X#5      [        S5      S
U" XC-  5      -  [        -   [#        XC-  5      -  S-  [        XC-  5      U " XC-  5      -  -
  -  [        [        5      -  :X  d   e[        [        S5      [%        [        R                  S4S4[        R                  SS4[        R                  44US-  US-  -  S-  5      -  S[        -  -  X25      SXB-   -  :X  d   e[        S[        US-  US-  -   5      -  X#5      [        S5      [%        [        R                  4S4S[        R                  44US-  US-  -  S-  5      -  S[        [        5      -  -  :X  d   e[        [        S5      [%        [        R                  4S4S[        R                  44US-  US-  -  S-  5      -  S[        [        5      -  -  X25      SU[        US-  US-  -  S-   5      -  -  :X  d   eg )Nr   )r~   r   r   r   r=   rC   rD   rQ   rg   rN   r   rd   )r   r   )r   r~   r   r   r   r	   r   r
   r   r&   r   r#   r   r[   r2   r(   r   r)   r3   )r~   r   r   r   r=   rC   s         rF   test_cosine_transformr      sj   @AAAA AaD!'?1Q4+FFFF#	!a-adA9: : : Ad1gIq,$q'	999#Ad1gIq4$q'	AAAA	1q!t#AwtBx/QBqD	91Q3?@ @ @ A	
1"qvv+&qq5z1%!a
2CCE!A#JNO O O#AQ	.. ) %a%j%r!taff}-%..3AaCj%9:;@CDr7K K K QBqD	1 GAItBxA1'=>? ? ?#Q	48QTAqD[)*A258!AY? ? ? C47
OC$	3 1 - QTE1Q3K((!Ax1~,=*=>? ? ? AquIq,Q	BqsGb#ac("1$s13x13'77199=b1B B B B#DGGaffa["4E	
A	H#$%qDAIaK-1 %123B$%89:?BCQU)L L L Ad1a4!Q$;//6$q''
&&B&166),ad1a4ikC; ;;<=d2hJ;H H H H#DGGaffYOfqvviEXZ[]^Z^_`bc_cZcdeZe,f$fhijnoqjrhr$suvz~  BC  DH  IJ  LM  IM  NO  QR  NR  IR  UV  IV  DW  BW  X  X  X  XrH   c            
      L   [        S5      n [        S5      n[        S5      n[        S5      n[        S5      n[        SU -  XS5      SU-  :X  d   e[        SU-  XS5      SU -  :X  d   e[        SX-  -  XS5      SU* S-   -  XS-
  -  -  [	        U* S-  S-   5      -  [	        US-  5      -  :X  d   e[        SU* S-   -  XS-
  -  -  [	        U* S-  S-   5      -  [	        US-  5      -  XS5      X* -  :X  d   e[        SX-  -  XU5      SSU* -  -  XS-
  -  -  [	        U* S-  US-  -   S-   5      -  [	        US-  US-  -   5      -  :X  d   e[        SU* S-   -  UUS-
  -  -  [	        U* S-  US-  -   S-   5      -  [	        US-  US-  -   5      -  XU5      X* -  :X  d   e[        X-  [        U* U -  5      -  XU5      SUS-   -  U-  X* S	-
  -  -  US-  US-  -  S-   U* [        S	5      S-  -
  -  -  [	        U[        S	S5      -   5      -  [        [        5      -  :X  d   e[        SUS-   -  U-  X* S	-
  -  -  US-  US-  -  S-   U* [        S	S5      -
  -  -  [	        U[        S	S5      -   5      -  [        [        5      -  XU5      X-  [        U* U -  5      -  :X  d   eg )
Nr   r   rl   mr=   rD   r   rQ   rg   )
r   r   r   r   r2   r#   r   r   r&   r   )r   r   rl   r   r=   s        rF   test_hankel_transformr   J  sb   sAsA	BsAAAaCq)QqS000#AaCq1QqS888	!$aQKE
25!A?B51:MN N N#	QBFAAJuaRTAX.uQqSz91DGH2wO O O AadFA"-	!qb'	!!e*UA2a4"Q$;?33E!A#1*4EEG G G#AQK	A1 %aRTBqD[1_%%&&+AaC"Q$J&7%89:rCFG"gN N N AE#qbd)OQ26	BFAa#'l"AqDAIMbS156<889<: 6: $; 	;;@hqRSnAT;U	VVZ[]V^	__ _ _ $	BFAa#'l"AqDAIMbS8Aq>5I#JJ5
Xa^L 	!"X	&'(R145E#qbd)OD D DrH   c                  N    [        SS[        -
  -  [        [        5      S :w  d   eg r   )r   r;   r<   r   rH   rF   test_issue_7181r   f  s!    Aq1uIq!,444rH   c                  v  ^  [         [        * S-   -  * SS[         S-  -  -   [        * S-  -  -  [        S[         S-  -  5      -  [        [        * [        -  [
        -  5      -  [        [        [        [        S[         S-  -  5      S-  5      -  5      -  [        [        5      -  m [        [        U 4S j5        g )NrD   rd   rQ   c                  D   > [        T [        [        S[        440 SSS.D6$ )NrY   T)
as_meijergneedeval)r   r<   r;   r   Fs   rF   <lambda>!test_issue_8882.<locals>.<lambda>v  s$     Aq2r( 	24
0	2rH   )r=   r<   r&   r#   r   r   r)   r'   r2   r:   r   r   s   @rF   test_issue_8882r   j  s     
aR!Va!AqD&jQBqD))$qAv,6sA2a47|CAd4!Q$<>""#	$$)!H	-A
! $2 3rH   c                  x    [        SSS9u  p[        [        U 5      X5      [        [        U 5      X5      :X  d   eg )Nzx yTr   )r   r   r#   r   )r;   ys     rF   test_issue_12591r   {  s5    5t$DASVQ*.>s1vq.LLLLrH   N)grE   r   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   sympy.integrals.laplacer   r   r   r   r   
sympy.corer   sympy.core.numbersr   r   r   r   sympy.core.singletonr   sympy.core.symbolr   r   (sympy.functions.combinatorial.factorialsr    $sympy.functions.elementary.complexesr!   r"   &sympy.functions.elementary.exponentialr#   r$   r%   rk   r&   r   r'   r(   r)   r*   sympy.functions.special.besselr+   r,   r-   r.   'sympy.functions.special.delta_functionsr/   r   r0   r1   'sympy.functions.special.gamma_functionsr2   sympy.functions.special.hyperr3   sympy.simplify.gammasimpr4   sympy.simplify.hyperexpandr5   sympy.simplify.trigsimpr6   sympy.testing.pytestr7   r8   r9   r:   	sympy.abcr;   r<   r=   r>   r?   r@   rl   rm   rn   rG   rK   rU   r_   rq   ru   r|   r   r   r   r   r   r   r   r   r   r   rH   rF   <module>r      s2  J J J J J2 4 ! 2 2 " - > ? F F 9 H H M M = ? 9 1 . 2 , : : & & &D#FA5& )  )8<F~A  [6 [6@ . .B JC JCZ2At RF'XTD853"MrH   